Transfer Function Modeling for the Buck converter - SPI · PDF file2 Transfer Function...

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Transfer Function Modelingfor the Buck converter

Sanda Lefteriu Cecile Labarre

Ecole des Mines de Douai

May 10, 2016

Sanda Lefteriu (Ecole des Mines de Douai) Transfer Function Modeling for the Buck converter May 10, 2016 1 / 20

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Motivation

Outline

1 Modeling Systems using Frequency Domain Measurements

2 Transfer Function Modeling for the Buck converterAverage model of the Buck converterGeneralized Transfer Function (GTF)Augmented MNA for PSL circuitsResults

3 Conclusion


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Motivation

Obtaining measurements in the frequency domain

Σu(t) = a0 sin(2πfi t + φ0) −→ −→ y(t) = a1 sin(2πfi t + φ1)

Problem statement: H(s) models a data set with N samples obtained from anunknown underlying system, if, for i=1, · · · ,N,

H(si = jωi ) ≈ Hi , where ωi = 2πfi ,Hi :=a1

a0φ1 − φ0.

For LTI systems, H(s) is a rational scalar/matrix function ⇒rational approximation problem.


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Modeling the Buck converter

Outline



3 Conclusion


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DC-DC switching power converters

They consist of LTI components (resistors, capacitors, inductors) together withswitches (transistors and diodes), whose operation is controlled to yield thedesired voltage conversion.

Frequency domain analysis via the Bode plot is not possible.

Ultimate goal: use the measurements to derive a model for the converter, validfrom low up to high frequency.


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State-of-the-art

Average models describe dynamics using average values of currents and voltages,neglecting ripple effects due to switching ⇒ state-space averaging (SSA)[Middlebrook & Cuk, 1976] employs the weighted combination of state equationsof switching phases in pulse-width modulated (PWM) converters. They are onlyvalid until half of the switching frequency.

In continuous current mode (CCM), the Buck converter has two working modes,so it can be modeled by the Generalized Transfer Function (GTF)[Biolek, 1997, Biolek et al., 2006], which combines the continuous and thediscrete behavior. The resulting function produces a Bode plot which is validpast half the switching frequency.

DC-DC power converters are periodic switched linear (PSL) circuits characterizedby time-varying transfer function or alternatively, by bi-frequency transferfunction [Zadeh, 1950]. The steady-state response found as solution ofaugmented MNA equation in the frequency domain [Trinchero, 2015].


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Modeling the Buck converter Average model of the Buck converter

Average model of the Buck converter

Figure : Topology of the Buck converter

In CCM, 2 linear working modes.1 1st mode: switch is closed (S = 1, S = 0) for t ∈ [kT , (k + D)T ], where D is

the duty cycle, T is the switching period and k = ...,−1, 0, 1, ...2 2nd mode: switch is open (S = 0, S = 1) for t ∈ [(k + D)T , (k + 1)T ].

Figure : Switching diagram


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Modeling the Buck converter Average model of the Buck converter

State-space equations

x(t) = Ax(t) + Biu(t), i = 1, 2, where (1)

x(t) =

[VC (t)IL(t)

],A =

[− 1

RC1C

− 1L 0

],B1 =

[01L

],B2 =

[00

],

with u(t) = Vin(t). Using the average values of VC and IL

˙x(t) = Ax(t) + Bu(t), where B =

[0DL

]. (2)

The output is y(t) = Vo(t) = VC (t) = Cx(t) + Du(t) with C =[1 0

], D = 0.

The line-to-output transfer function is H(s) = Vo(s)Vin(s) . Its expression based on

average model is

Havg (s) =D

1 + s LR + s2LC

. (3)


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Modeling the Buck converter Generalized Transfer Function (GTF)

Deriving the GTF

We follow the derivation in [Biolek, 1997, Biolek et al., 2006]. The solution of (1):

x(t) = eA(t−t0)x0 +

∫ t

t0

eA(t−τ)Biu(τ)dτ, t ≥ t0. (4)

Phase 1: t ∈ [kT , kT + DT ), write (4) at the end of it

x(kT + DT ) = eADTx(kT ) +

∫ kT+DT

kT

eA(kT+DT−τ)B1u(τ)dτ. (5)

Phase 2: t ∈ [kT + DT , (k + 1)T ), write (4)

x(t) = eA(t−kT−DT )x(kT + DT ). (6)

We replace (5) into (6) and evaluate at the end of the period, for t = (k + 1)T :

x((k + 1)T )=eATx(kT ) + eA(k+1)T

∫ kT+DT

kT

e−AτB1u(τ)dτ.


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Deriving the GTF

Buck converter is a DC-DC converter ⇒ use small signal analysis:u(t) = u0 + ue jΩt , with u0 >> u and Ω, perturbation frequency of choice.

The output at steady-state contains a DC component and an AC of the samefrequency Ω, together with other frequencies. Using the analogy with LTI, onlythe component with frequency Ω is of interest.The generalized transfer function for the line-to-output response is given by

HGTF (jΩ) = C(e jΩT I−eAT

)−1eAT

∫ DT

0

e−AτB1ejΩτdτ, (7)

where I is the identity matrix.


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Deriving the GTF

Buck converter is a DC-DC converter ⇒ use small signal analysis:u(t) = u0 + ue jΩt , with u0 >> u and Ω, perturbation frequency of choice.The output at steady-state contains a DC component and an AC of the samefrequency Ω, together with other frequencies. Using the analogy with LTI, onlythe component with frequency Ω is of interest.

The generalized transfer function for the line-to-output response is given by


)−1eAT

∫ DT

0




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Deriving the GTF

Buck converter is a DC-DC converter ⇒ use small signal analysis:u(t) = u0 + ue jΩt , with u0 >> u and Ω, perturbation frequency of choice.The output at steady-state contains a DC component and an AC of the samefrequency Ω, together with other frequencies. Using the analogy with LTI, onlythe component with frequency Ω is of interest.The generalized transfer function for the line-to-output response is given by


)−1eAT

∫ DT

0




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Modeling the Buck converter Augmented MNA for PSL circuits

Time-varying systems

Following [Zadeh, 1950], impulse response and transfer function can be extendedto time-varying systems. The output is

y(t) =

∫ ∞−∞

h(t, τ)u(τ)dτ, h(t, τ) = R[δ(t − τ)],

with h(t, τ), the generalized impulse response, R, the operator describing systembehavior, t, the observation time and τ , the excitation time.The bi-frequency transfer function is

H(ω,Ω) =

∫ ∞−∞

∫ ∞−∞

h(t, τ)e−j(ωt−Ωτ)dtdτ. (8)

where Ω and ω are the input and output frequencies.The time-varying transfer function

H(t,Ω) =

∫ ∞−∞

h(t, τ)e−jΩ(t−τ)dτ. (9)


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Periodical linear time-varying (PLTV) systems

H(t,Ω) is a periodic function of t ⇒ Fourier series w.r.t. ωs = 2πT :

H(t,Ω) =n=∞∑n=−∞

Hn(Ω)e jnωs t . (10)

Frequency-dependent Fourier coefficients Hn(Ω) called aliasing transfer functions:

Hn(Ω) =1

T

∫ T

0

H(t,Ω)e−jnωs tdt.

The output in the frequency domain:

Y (ω) =n=∞∑n=−∞

Hn(ω − nωs)U(ω − nωs). (11)

The LTI relationship can be recovered for n = 0, namely Y (ω) = Hn(ω)U(ω).


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Periodical linear time-varying (PLTV) systems

Figure : Block-diagram representation with infinitely many LTI systems [Trinchero, 2015]

Figure : Output for single tone ω0 contains new harmonics at multiples of ωs


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Modified Nodal Analysis (MNA) for PSL circuits

Approach from [Trinchero, 2015] for computing steady-state of periodic switchedlinear circuits via augmented time-invariant MNA equation in frequency-domainadapted for small-signal transfer function computation.

Figure : Frequency domain representation of the Buck converter [Trinchero, 2015]

Substitute S , S with periodic switching elements Y1, Y2 and replace voltagesource by its Norton equivalent ⇒ the augmented MNA equation isY1 + Y2 0 I

0 jωωωC + G I −II −I −jωωωL

V1

V2

IL

=

Y1Vin

00

, (12)

where G = 1R and each block is a matrix of dimension 2N + 1.


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MNA for PSL circuits

IL contains coefficients of truncated frequency domain representation

IL(ω) =N∑

n=−N

Inδ(ω − nωs − ω0), (13)

with ω0, the excitation frequency. Similarly for V1, V2. Y1 and Y2 are Toeplitz

Yi =

Yi,0 Yi,−1 . . . Yi,−2N

Yi,1 Yi,0 . . . Yi,−2N+1

......

. . ....

Yi,2N Yi,2N−1 . . . Yi,0

, i = 1, 2, (14)

Y1,n are Fourier coefficients of window function with amplitude 1 when S is on:

Y1,n = fs

∫ DT

0

e−jnωs tdt =e−jnωsDT − 1

−j2πn, n = −2N, . . . , 2N,

in Y1, and Y2,n = Y1ne−jωsnDT , m=−2N, . . . , 2N in Y2, for S due to delay DT .

ωωω = diag([

ω0 − Nωs . . . ω0 + Nωs

]).


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MNA for PSL circuits

Small-signal analysis: u(t) = u0 + ue jΩt , with u0 >> u and Ω, perturbationfrequency ⇒ U(ω) = u0δ(ω) + uδ(ω − Ω). Linear system solved for each input

UDC =

0u0

0

, UAC =

0u0

, 0 ∈ Rn×1.

We have V2 = V2,DC + V2,AC , where

V2,DC (ω) =N∑

n=−N

Vn,DCδ(ω − nωs), (15)

V2,AC (ω) =N∑

n=−N

Vn,ACδ(ω − nωs − Ω), (16)

with coefficients Vn,DC , Vn,AC found by solving (12) for ω0 = 0, ω0 = Ω. Thetransfer function at perturbation frequency Ω is

HMNA(jΩ) =

V0,AC

u , if Ω 6= nωsVn,DC +Vn,AC

u , if Ω = nωs

. (17)

This expression clearly shows that if the perturbation frequency Ω is an integermultiple of the switching frequency ωs , there are components due to the DC inputwhich also contribute to the response.


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Modeling the Buck converter Results

Simulation and Experimental Results

Buck converter with parameters: L = 1mH, C = 500µF, R = 12Ω,fs = 20kHz= 1

T , duty cycle D = 12 .

Figure : Simulation results and experimental validationSanda Lefteriu (Ecole des Mines de Douai) Transfer Function Modeling for the Buck converter May 10, 2016 17 / 20

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Conclusion

Outline



3 Conclusion


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Conclusion

Conclusion & Future work

Presented:

The line-to-output transfer function was simulated using the average model, theGTF and the MNA for PSL circuits.

Its magnitude was measured with a spectrum analyzer.

Next steps:

Consider other modeling techniques (e.g., take into account parasitics).

Measure the phase.

Model, simulate and measure the control-to-output transfer function.

Thank you for your attention!


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Conclusion

References

Biolek, D. (1997).

Modeling of periodically switched networks by mixed s-z description.IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 44(8), 750–758.

Biolek, D., Biolkova, V., & Dobes, J. (2006).

Modeling of switched DC-DC converters by mixed s-z description.In Proceedings of the IEEE International Symposium on Circuits and Systems (pp. 831–834).

Middlebrook, R. & Cuk, S. (1976).

A general unified approach to modelling switching-converter power stages.In IEEE Power Electronics Specialists Conference (pp. 18–34).

Trinchero, R. (2015).

EMI Analysis and Modeling of Switching Circuits.PhD thesis, Politecnico di Torino.

Zadeh, L. A. (1950).

Frequency analysis of variable networks.Proceedings of the IRE, 38(3), 291–299.


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